You aren't doing your baby a disservice by getting out, but it would probably be a good thing for you, for you to get out more.
Before you know it, outings will be all soft play, playdates and parks. This first few months is the time when you can go places for YOU.
Places you've never had time to go to before because of work, long lunches with friends, walks, museums, whatever you want really. And baby wont mind either way as they just need you near at that stage, wherever that is.
But if you'd rather be at home, then that's fine too.
Honestly, no one cares if baby cries when you are out, though I can understand feeling self conscious (and did myself). At this stage, they are just a baby, and people expect them to cry sometimes. At worst, the noise may irritate people, but they wont be judging you or your baby.
Also (Im sorry) but learning how to cope with being embarrassed in public is something you cant wait out, as it goes on for many years.
My excruciation at not being able to get my baby to latch on in a cafe is nothing in hindsight, compared to a 2yo lying on the floor of a supermarket screaming at me, whilst calling me names. Or a 3yo loudly saying out of the blue when in a queue, how they 'loved their daddy's penis'.
Motherhood is amazing, but its a big learning curve. All your baby needs is you right now, and you are doing a fab job with that.